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Long time evolution of a pair of 2D viscous point vortices

Ping Zhang, Yibin Zhang

Abstract

This paper studies the long-time evolution of two point vortices under the 2D Navier-Stokes tokes equations. Starting from initial data given by a pair of Dirac measures, we derive an asymptotic expansion for the vorticity over time scales significantly longer than the advection time, yet shorter than the diffusion time. Building on previous works \cite{GS24-1, DG24}, we construct suitable approximate solutions $Ω_a$ and employ Arnold's method to define a nonlinear energy functional $E_\ve[\om]$, with respect to which the linearized operator $Λ^{E,\star}$ around $Ω_a$ is nearly skew-adjoint. A key innovation in this work is the introduction of ``pseudo-momenta'': $\varrho^e_a, \varrho^o_a,\varrho^{te}_a, \varrho^{to}_a$, which correspond to eigenfunctions or other nontrivial elements in invariant subspaces of $Λ^E$, derived from the Lie structure of the 2D Euler equations.

Long time evolution of a pair of 2D viscous point vortices

Abstract

This paper studies the long-time evolution of two point vortices under the 2D Navier-Stokes tokes equations. Starting from initial data given by a pair of Dirac measures, we derive an asymptotic expansion for the vorticity over time scales significantly longer than the advection time, yet shorter than the diffusion time. Building on previous works \cite{GS24-1, DG24}, we construct suitable approximate solutions and employ Arnold's method to define a nonlinear energy functional , with respect to which the linearized operator around is nearly skew-adjoint. A key innovation in this work is the introduction of ``pseudo-momenta'': , which correspond to eigenfunctions or other nontrivial elements in invariant subspaces of , derived from the Lie structure of the 2D Euler equations.

Paper Structure

This paper contains 33 sections, 31 theorems, 312 equations, 1 figure.

Table of Contents

  1. introduction
  2. Well-posedness of \ref{['eq 1.1']} and enhanced dissipation of Oseen vortices
  3. Point vortex system
  4. Previous studies on the long time evolution of Dirac measures
  5. Main results.
  6. Sketch of the proof and new ingredients.
  7. Notations:
  8. Preliminaries
  9. The rescaled vorticity equations
  10. Basic conservation
  11. Basic operators
  12. Expansion of the interacting velocity
  13. The construction of approximate solutions
  14. Analysis of the momentum.
  15. Construction of approximate solutions
  16. ...and 18 more sections

Key Result

Theorem 1.1

For any $w_0\in L^1(\Bbb R^2)$, the solution $w(t,x)$ of eq 1.1 satisfies where $\Gamma\buildrel\hbox{\footnotesize def}\over = \int_{\Bbb R^2} w_0(x) dx$.

Figures (1)

  • Figure 1: In the inner region $\mathrm{I}_{i,\varepsilon}$, the weight is close for $\varepsilon > 0$ small to the radially symmetric function $W_0(\xi) = 4|\xi|^{-2} \bigl(e^{|\xi|^2/4}-1\bigr)$. It then takes constant values in the intermediate region $\mathrm{II}_{i,\varepsilon}$, and grows like $\exp(|\xi|^{2\gamma}/4)$ in the outer region $\mathrm{III}_{i,\varepsilon}$. The dashed lines illustrate the bounds \ref{['eq 5.8a']}, where the constants $C_1, C_2$ are independent of $\varepsilon$. This picture is borrowed from DG24.

Theorems & Definitions (61)

  • Theorem 1.1: GW05
  • Theorem 1.2: GW05
  • Theorem 1.3: G12GR08
  • Theorem 1.4
  • Theorem 1.5: DG24
  • Theorem 1.6: G11
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 51 more